Exploring Triangle Similarity: AA, SSS, and SAS Methods
AA,SSS and SAS similarity
Warm Up
Solve each proportion.
1.
2.
3.
4.
If
∆
QRS
~
∆
XYZ
, identify the pairs of congruent
angles and write 3 proportions using pairs of
corresponding sides.
z
=
±
10
x
= 8
Q
X;
R
Y;
S
Z;
Prove certain triangles are similar by
using AA, SSS, and SAS.
Use triangle similarity to solve problems.
Objectives
There are several ways to prove certain triangles are
similar. The following postulate, as well as the SSS
and SAS Similarity Theorems, will be used in proofs
just as SSS, SAS, ASA, HL, and AAS were used to
prove triangles congruent.
Example 1: Using the AA Similarity Postulate
Explain why the triangles
are similar and write a
similarity statement.
Check It Out!
Example 1
Explain why the triangles
are similar and write a
similarity statement.
By the Triangle Sum Theorem, m
C
= 47
°
, so
C
F
.
B
E
by the Right Angle Congruence Theorem.
Therefore,
∆
ABC
~
∆
DEF
by AA ~.
Example 2A: Verifying Triangle Similarity
Verify that the triangles are similar.
∆
PQR
and
∆
STU
Therefore
∆
PQR
~
∆
STU
by SSS
~
.
Example 2B: Verifying Triangle Similarity
∆
DEF
and
∆
HJK
Verify that the triangles are similar.
D
H
by the Definition of Congruent Angles.
Therefore
∆
DEF
~
∆
HJK
by SAS ~.
Check It Out!
Example 2
Verify that
∆
TXU
~
∆
VXW
.
TXU
VXW
by the
Vertical
Angles
Theorem.
Therefore
∆
TXU
~
∆
VXW
by SAS ~.
A
A
by Reflexive Property of
, and
B
C
since they are both right angles.
Example 3: Finding Lengths in Similar Triangles
Explain why
∆
ABE
~
∆
ACD
, and
then find
CD
.
Step 1
Prove triangles are similar.
Therefore
∆
ABE
~
∆
ACD
by AA ~.
Example 3 Continued
Step 2
Find
CD
.
Corr. sides are proportional.
Seg. Add. Postulate.
Substitute x for CD, 5 for BE,
3 for CB, and 9 for BA.
Cross Products Prop.
x
(9) = 5(3 + 9)
Simplify.
9
x
= 60
Divide both sides by 9.
Check It Out!
Example 3
Explain why
∆
RSV
~
∆
RTU
and then find
RT
.
Step 1
Prove triangles are similar.
It is given that
S
T
.
R
R
by Reflexive Property of
.
Therefore
∆
RSV
~
∆
RTU
by AA ~.
Check It Out!
Example 3 Continued
Step 2
Find
RT
.
Corr. sides are proportional.
Substitute RS for 10, 12 for
TU, 8 for SV.
Cross Products Prop.
Simplify.
Divide both sides by 8.
RT
(8) = 10(12)
8
RT
= 120
RT
= 15
Example 4: Writing Proofs with Similar Triangles
Given: 3
UT
= 5
RT
and 3
VT
= 5
ST
Prove:
∆
UVT
~
∆
RST
1.
Given
1.
3
UT
= 5
RT
2.
Divide both sides by 3
RT
.
3.
Given.
3.
3
VT
= 5
ST
4.
Divide both sides by3
ST
.
5.
Vert.
s Thm.
5.
RT
S
V
TU
6.
SAS ~
Steps 2, 4, 5
6.
∆
UVT
~
∆
RST
Example 4 Continued
Check It Out!
Example 4
Check It Out!
Example 4 Continued
1.
Given
2.
∆
Midsegs. Thm
3.
Div. Prop. of =.
4.
SSS ~
Step 3
4.
∆
JKL
~
∆
NPM
Example 5: Engineering Application
From p. 473, BF
4.6 ft.
BA
=
BF + FA
6.3 + 17
23.3 ft
Therefore,
BA
= 23.3 ft.
Check It Out!
Example 5
What if…?
If
AB
= 4
x
,
AC
= 5
x
, and
BF
= 4, find
FG
.
Corr. sides are proportional.
Substitute given quantities.
Cross Prod. Prop.
Simplify.
4
x
(
FG
) = 4(5
x
)
FG
= 5
You learned in Chapter 2 that the Reflexive,
Symmetric, and Transitive Properties of Equality
have corresponding properties of congruence.
These properties also hold true for similarity of
triangles.
Lesson Quiz
1.
Explain why the triangles are
similar and write a similarity
statement.
2.
Explain why the triangles are
similar, then find
BE
and
CD
.
Lesson Quiz
1. By the Isosc.
∆
Thm.,
A
C,
so by the def.
of
, m
C
=
m
A
. Thus m
C
=
70
°
by subst.
By the
∆
Sum Thm., m
B
=
40
°
. Apply the
Isosc.
∆
Thm. and the
∆
Sum Thm. to
∆
PQR
.
m
R
= m
P
=
70
°
. So by the def. of
,
A
P,
and
C
R
. Therefore
∆
ABC
~
∆
PQR
by AA
~
.
2.
A
A
by the Reflex. Prop. of
. Since
BE
||
CD
,
ABE
ACD
by the Corr.
s Post.
Therefore
∆
ABE
~
∆
ACD
by AA ~.
BE
= 4 and
CD
= 10.
Delve into the concepts of triangle similarity using the AA, SSS, and SAS methods. Understand how to verify and prove triangles are similar, identify congruent angles, and write proportions. Discover examples illustrating these principles through clear explanations and visual aids.
Uploaded on Sep 30, 2024 | 0 Views
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
Presentation Transcript
Warm Up Solve each proportion. 1. 2. 3. x = 8 z = 10 4. If QRS ~ XYZ, identify the pairs of congruent angles and write 3 proportions using pairs of corresponding sides. Q X; R Y; S Z;
Objectives Prove certain triangles are similar by using AA, SSS, and SAS. Use triangle similarity to solve problems.
There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.
Example 1: Using the AA Similarity Postulate Explain why the triangles are similar and write a similarity statement. Since , B E by the Alternate Interior Angles Theorem. Also, A D by the Right Angle Congruence Theorem. Therefore ABC ~ DEC by AA~.
Check It Out! Example 1 Explain why the triangles are similar and write a similarity statement. By the Triangle Sum Theorem, m C = 47 , so C F. B E by the Right Angle Congruence Theorem. Therefore, ABC ~ DEF by AA ~.
Example 2A: Verifying Triangle Similarity Verify that the triangles are similar. PQR and STU Therefore PQR ~ STU by SSS ~.
Example 2B: Verifying Triangle Similarity Verify that the triangles are similar. DEF and HJK D H by the Definition of Congruent Angles. Therefore DEF ~ HJK by SAS ~.
Check It Out! Example 2 Verify that TXU ~ VXW. TXU VXW by the Vertical Angles Theorem. Therefore TXU ~ VXW by SAS ~.
Example 3: Finding Lengths in Similar Triangles Explain why ABE ~ ACD, and then find CD. Step 1 Prove triangles are similar. A A by Reflexive Property of , and B C since they are both right angles. Therefore ABE ~ ACD by AA ~.
Example 3 Continued Step 2 Find CD. Corr. sides are proportional. Seg. Add. Postulate. Substitute x for CD, 5 for BE, 3 for CB, and 9 for BA. Cross Products Prop. x(9) = 5(3 + 9) 9x = 60 Simplify. Divide both sides by 9.
Check It Out! Example 3 Explain why RSV ~ RTU and then find RT. Step 1 Prove triangles are similar. It is given that S T. R R by Reflexive Property of . Therefore RSV ~ RTU by AA ~.
Check It Out! Example 3 Continued Step 2 Find RT. Corr. sides are proportional. Substitute RS for 10, 12 for TU, 8 for SV. RT(8) = 10(12) Cross Products Prop. 8RT = 120 Simplify. RT = 15 Divide both sides by 8.
Example 4: Writing Proofs with Similar Triangles Given: 3UT = 5RT and 3VT = 5ST Prove: UVT ~ RST
Example 4 Continued Statements Reasons 1. Given 1. 3UT = 5RT 2. 2. Divide both sides by 3RT. 3. 3VT = 5ST 3. Given. 4. Divide both sides by3ST. 4. 5. Vert. s Thm. 5. RTS VTU 6. UVT ~ RST 6. SAS ~ Steps 2, 4, 5
Check It Out! Example 4 Given: M is the midpoint of JK. N is the midpoint of KL, and P is the midpoint of JL.
Check It Out! Example 4 Continued Statements Reasons 1. Given 1. M is the mdpt. of JK, N is the mdpt. of KL, and P is the mdpt. of JL. 2. 2. Midsegs. Thm 3. 3. Div. Prop. of =. 4. JKL ~ NPM 4. SSS ~ Step 3
Example 5: Engineering Application The photo shows a gable roof. AC || FG. ABC ~ FBG. Find BA to the nearest tenth of a foot. From p. 473, BF 4.6 ft. BA = BF + FA 6.3 + 17 23.3 ft Therefore, BA = 23.3 ft.
Check It Out! Example 5 What if ? If AB = 4x, AC = 5x, and BF = 4, find FG. Corr. sides are proportional. Substitute given quantities. 4x(FG) = 4(5x) Cross Prod. Prop. Simplify. FG = 5
You learned in Chapter 2 that the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. These properties also hold true for similarity of triangles.
Lesson Quiz 1. Explain why the triangles are similar and write a similarity statement. 2. Explain why the triangles are similar, then find BE and CD.
Lesson Quiz 1. By the Isosc. Thm., A C, so by the def. of , m C = m A. Thus m C = 70 by subst. By the Sum Thm., m B = 40 . Apply the Isosc. Thm. and the Sum Thm. to PQR. m R = m P = 70 . So by the def. of , A P, and C R. Therefore ABC ~ PQR by AA ~. 2. A A by the Reflex. Prop. of . Since BE || CD, ABE ACD by the Corr. s Post. Therefore ABE ~ ACD by AA ~. BE = 4 and CD = 10.
